Method of measuring the effective directivity and/or residual system port impedance match of a system-calibrated vector network analyser

ABSTRACT

To measure the effective directivity and/or the effective source port match of a test port of a system-calibrated vector network analyser, a precision air line short-circuited at the outlet is connected, and the complex reflection coefficient is measured at the inlet of this precision air line at a sequence of measuring points within a predefined frequency range. At the same time, for the effective directivity the sequence of the measured complex reflection coefficients is subjected to a discrete Fourier transformation and the baseband filtered out of the spectrum thereby formed. The sequence of effective directivity values is obtained by subsequent inverse Fourier retransformation.

The invention relates to a method of measuring the effective directivity(residual system directivity) and/or the effective source port match(residual system port impedance match) of a test port of asystem-calibrated vector network analyser in accordance with thepreamble of claim 1.

The great accuracy of vector network analysers (VNA) is based on thefact that, before the actual measurement of value and phase of thecomplex reflection coefficient, the network analyser is calibrated atits test ports by connecting calibration standards. Meanwhile there is alarge number of different calibration methods. For system calibration,the most common of these use open-circuit, short-circuit and matchcalibration standards. By connecting these calibration standards to thetest ports of the network analyser, it is possible to determine theerrors occurring in the network analyser which lead to a deviation ofthe measured values from the true value, and this information may thenbe used in the subsequent object measurement for error correction bycalculation. This is known for example from DE 39 12 795 A1. However,these calibration methods as commonly used to date are not sufficientlyaccurate.

In order to determine the still remaining residual uncertainty of thedirectivity and/or test port match, it is proposed in an EA guidelinethat a falsely-terminated or short-circuited precision coaxial air line,defined at the outlet, be connected to the test port to be measured ofthe previously system-calibrated network analyser, and that thereflection coefficients be measured at the inlet to this air line, at asequence of measuring points within a predefined frequency range of thenetwork analyser (EA-10/12, EA Guidelines on the Evaluation of VectorNetwork Analysers (VNA), European Co-operation for Accreditation, May2000). According to this guideline, though, only the so-called rippleamplitude of the oscillation overlying the value of the reflectioncoefficients is evaluated, and it is assumed as a simplification thatthis ripple amplitude is broadly identical to the effective source portmatch, which however is true only if the effective directivity isignored. This known verification standard using a precision air line istherefore relatively inaccurate and allows no precise estimate of themeasuring uncertainty to be expected, let alone any subsequentcorrection of the error correction terms for the source port match.

The problem of the invention is to indicate a method of measurement andto create a set of calibration standards with which the effectivedirectivity and/or effective source port match may be determined withsubstantially greater accuracy, and specifically with such accuracy thateven the error correction values determined during system calibrationand stored in the network analyser may be suitably re-corrected.

This is solved by a method according to claim 1 or claim 2 and a set ofcalibration standards according to claim 10.

Using the method according to the invention, the measured complexreflection coefficients may be used to determine the effectivedirectivity and/or the effective source port match with substantiallygreater accuracy than is possible using the known EA guideline. Thedetermined measured values are available with a level of accuracycorresponding to that of the precision air line used. Here it is notimportant if the impedance of the precision air line used deviates fromthe reference impedance of the network analyser since, according to theinvention, even such impedance variations may be taken into account, solong as they are known. This correction is also possible even down torelatively low frequencies, at which the impedance of the air linediffers increasingly from the nominal value due to the reducing skineffect. Since a variation in impedance of the air line is acceptableunder the method according to the invention, less costly air lines mayalso be used for the measurement, so long as adequate longitudinalhomogeneity of the cross-sectional dimensions is ensured. Moreover, inprinciple it is only necessary to measure with a short-circuitedprecision air line, whereas using the known method it is still essentialto make an additional measurement with a defined false termination. Theadditional measurement with a defined false termination of an air lineas provided under the invention may be used to cover operating errors.By this means, routine checking and determination of the residual errorin network analysers is considerably simplified.

A special advantage of the method according to the invention is that,with the achievable accuracy in measurement of the effective source portmatch and/or the effective directivity, it is possible to correct theactual error correction values obtained through the precedingcalibration of the network analyser and stored in the latter. In thisway, the measuring accuracy of any such vector network analyser issubstantially enhanced and a measuring accuracy is obtained whichmatches the quality of the precision air line. The method according tothe invention may be used both for network analysers with only one testport (reflectometer) and also for those with two or more test ports. Inthe case of several test ports, the measurement of effective directivityand effective source port match is made, in accordance with theinvention, separately at each of the test ports.

Since the values for effective directivity and effective source portmatch measured by the method according to the invention as a function offrequency are essentially to be included with the calibration standardsused in system calibration of the network analyser, it makes sense tostore these measured residual error values on a suitable data medium,for example in the form of measurement records or diagrams or as digitalvalues on a diskette, and to add them to the calibration kit to be usedin system calibration, so that the user, after calibrating his networkanalyser with the calibration standards, will immediately input thedetermined residual error values to the network analyser so that thecalibration data stored there may be suitably corrected.

The invention will be explained in detail below with the aid ofschematic drawings and digrams relating to a mathematical model. Thedrawings show as:

FIG. 1 the signal-flow graphs of a VNA for impedance measurement;

FIG. 2 the signal-flow graphs of a falsely-terminated line;

FIG. 3 the equivalent network diagram of a falsely-terminated air line;

FIG. 4 measured values for the value of the reflection coefficient atthe inlet of the short-circuited air line;

FIG. 5 measured values for the complex reflection coefficients at theinlet of the short-circuited air line in a parametric representation;

FIG. 6 the discrete Fourier transformation (DFT) of the reflectioncoefficient at the inlet of the short-circuited air line;

FIG. 7 extended and reflected point sequence;

FIG. 8 filtering-out of the mixed-down carrier with low-pass transferfunction;

FIG. 9 the values of the reflection coefficient and the carriercomponent in comparison;

FIG. 10 the value of the sideband signal;

FIG. 11 the spectrum of the sideband signal with low-pass transferfunction;

FIG. 12 the component B_(n) in polar coordinate display format(parameter: frequency in GHz). The right-hand chart is obtained with agreatly reduced bandwidth of the low-pass filter (1/40);

FIG. 13 the spectrum of the term S_(ac)/A² with low-ass transferfunction;

FIG. 14 the component C_(n)/A_(n) ² in polar coordinate display format(parameter: frequency in GHz). The right-hand chart is obtained with agreatly reduced bandwidth of the low-pass filter (1/5);

FIG. 15 the vector difference of the measured result according to FIG.14;

FIG. 16 the ripple amplitude of the measured result according to FIG. 4in comparison with the values of the measured results for “residualsystem directivity” and “residual system port impedance match” using themethod according to the invention (f/GHz=n/10);

FIG. 17 a section of the spectrum of the mixed-down signal according toFIG. 8, shown for different degrees of extrapolation. I: upperextrapolation, upper reflection, 400 points. II: 1600 pointscorresponding to FIG. 7 (8). III: correspond to FIG. 18.

FIG. 18 the point sequence with measuring points from FIG. 1, greatlyextended and led to locate extremes, and

FIG. 19 the basic structure of a network analyser.

FIG. 19 shows the basic structure of a network analyser N, andspecifically for the sake of simplicity a network analyser N with onlyone test port M (reflectometer). An integral generator G feeds the testport M (outer test port n comparison with FIG. 1) with a measuringsignal of variable frequency and specifically for example in equidistantfrequency steps between 100 MHz and 40 GHz. Inserted in the connectingline between generator G and test port M are two directional couplers R1and R2. The amplitude of the outgoing wave a is measured by R1 and theamplitude of the reflected wave by the other unit R2. In order tocalibrate a network analyser of this kind, various calibration standardssuch as open-circuit, short-circuit and match are connected insuccession to the test port M and used to carry out calibrationmeasurements. The error values thus determined are combined at an errortwo-input port Z and for example stored as error correction values in amicroprocessor X in the network analyser.

To estimate the accuracy with which a network analyser of this kind issystem-calibrated, the invention states that a coaxial precision airline L of a prescribed minimum length is connected at the test port M,and is short-circuited at its outlet which faces away from the test portM. This air line may now be used to measure the complex reflectioncoefficient in the frequency range of the generator G at a succession ofequidistant measuring points, from which the effective directivityand/or the effective source port match are determined in accordance withthe mathematical model below. These may then be used to correct in turnthe error correction values of the network analyser stored in themicroprocessor X.

The following symbols are used in the mathematical model below:

a_(e)b_(e) prediction coefficients

A_((n)) “carrier” signal within the recorded set of measured values (nthmeasuring point)

A′_((n)) mixed-down “carrier” (nth measuring point)

α damping constant of the reference air line

B_((n)) “baseband” signal within the recorded set of measured values(nth measuring point)

β phase constant of the reference air line

cfft(x) discrete (complex) Fourier transformation

C_((n) signal for the doubled carrier frequency within the recorded set of measured values (nth measuring point))

δ residual system directivity (effective directivity)

Δ error vector magnitude (value of the error vector)

ΔL₁₍₂₎ equivalent inductivity for the influence of the male connector atport 1 (2) of the reference air line

ΔC₁₍₂₎ equivalent capacity for the influence of the male connector atport 1 (2) of the reference air line

ΔX₁₍₂₎ reactance in series to the male connector 1 (2) of the referenceair line

ΔY₁₍₂₎ susceptance parallel to the male connector 1 (2) of the referenceair line

e index

E_(a(e)) number of points added by linear prediction at the start (a) orfinish (e) of the sequence of measuring points

F “frequency” of the “carrier” signal

γ propagation constant of the reference air line

Γ_(a) reflection coefficient of a measured object

Γ_(ac) reflection coefficient Γ_(a) (after system error correction)

Γ_(nc) nth measuring point for Γ_(ac)

Γ_(nc,mix) mixed-down sequence (nth measuring point)

Γ₁ reflection coefficient of the false termination at the outlet of thereference air line

κ proportionality constant

k index

k_(max) maximum value of k

l overall length of the line between the reference plane of the VNA andthe plane of the false termination

l₁ the length of the reference air line between the reference planes ofthe two male connectors

l₂ length of the line section in the physical component “falsetermination”

μ residual system port impedance match (effective source port match)

n, n_(a), n_(e) indices

N number of measuring points

υ index

p index

P number of prediction coefficients

r_(s) standardised impedance deviation of the reference air line(equivalent reflection factor)

s_(xy) s-parameter of the reference air line

S_(ac) _(n) sideband signal (nth measuring point)

t time

t₁₍₂₎ equivalent time constants for the male connector influence at port1 (2) of the reference air line

T estimated value for the reflection tracking

τ residual system tracking

ω circuit frequency

Z₀ reference impedance of the VNA

ΔZ impedance deviation of the reference air line

Starting from FIG. 1Γ_(ac)≅δ+(1+τ)Γ_(x)+μΓ_(a) ²   (1)applies for the reflection coefficients of a measured object correctedwith allowance for the error terms.

According to FIG. 2 $\begin{matrix}{\Gamma_{a} = {s_{11} + {\frac{s_{12}s_{21}}{1 - {s_{22} \cdot \Gamma_{1}}} \cdot \Gamma_{1}}}} & (2)\end{matrix}$applies for the inlet-side reflection coefficients of an air linefalsely terminated with Γ₁. Ignoring products of the terms δ, τ, μ, s₁₁and s₂₂ together, we obtain:Γ₂₀ ≅δ+s ₁₁+(1+τ+s _(ss)Γ₁)s ₁₂ s ₂₁Γ₁ +μs ₁₂ ² s ₂₁ ²Γ₁ ²   (3).

To determine the s-parameter of the air line, use is made of theequivalent network diagram in FIG. 3. The homogeneous part of the airline between the two connectors has here the length l₁ and ischaracterised by the propagation constant γ=α+β and an impedancedeviation ΔZ. The short line with length l₂ is part of the falsetermination with the reflection coefficient Γ₁, and specifically thesection between the reference plane and the plane of the falsetermination. The impedance deviation of this section is assumed to bezero, and the propagation constant as great as that of the longhomogeneous section of line. Any impedance deviations may be added inthe form of reactances to the half-section between the two sections ofline. The two half-sections represent the interference points formed bythe HF connectors.l=l ₁ +l ₂   (4)s₁₂=s₂₁≅e^(−γ1)   (5)applies as a simple approximation.

For the reflection parameters $\begin{matrix}\begin{matrix}{s_{11} = {\frac{{\Delta\quad Z} + {j\quad\Delta\quad X_{1}}}{2Z_{0}} - {j\frac{\Delta\quad Y_{1}Z_{0}}{2}} -}} \\{\left\lbrack {\frac{{\Delta\quad Z} - {j\quad\Delta\quad X_{2}}}{2Z_{0}} + {j\frac{\Delta\quad Y_{2}Z_{0}}{2}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l_{2}}} \\{= {\frac{{\Delta\quad Z} + {j\quad\Delta\quad X_{1}}}{2Z_{0}} - {j\frac{\Delta\quad Y_{1}Z_{0}}{2}} -}} \\{\left\lbrack {\left\lbrack {\frac{{\Delta\quad Z} - {j\quad\Delta\quad X_{2}}}{2Z_{0}} + {j\frac{\Delta\quad Y_{2}Z_{0}}{2}}} \right\rbrack{\mathbb{e}}^{2\gamma\quad l_{2}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l}}\end{matrix} & (6) \\\begin{matrix}{s_{22} = \left\lbrack {\frac{{\Delta\quad Z} + {j\quad\Delta\quad X_{2}}}{2Z_{0}} - {j\frac{\Delta\quad Y_{2}Z_{0}}{2}} -} \right.} \\{\left. {\left\lbrack {\frac{{\Delta\quad Z} - {j\quad\Delta\quad X_{1}}}{2Z_{0}} + {j\frac{\Delta\quad Y_{1}Z_{0}}{2}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l_{1}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l_{2}}} \\{= \left\lbrack {\frac{{\Delta\quad Z} + {j\quad\Delta\quad X_{2}}}{2Z_{0}} - {j\frac{\Delta\quad Y_{2}Z_{0}}{2}}} \right\rbrack} \\{{\mathbb{e}}^{{- 2}\gamma\quad l_{2}} - {\left\lbrack {\frac{{\Delta\quad Z} - {j\quad\Delta\quad X_{3}}}{2Z_{0}} + {j\frac{\Delta\quad Y_{1}Z_{0}}{2}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l}}}\end{matrix} & (7)\end{matrix}$apply.Case 1: Γ₁=−1 (Verification With Outlet-Side Short-Circuit)

After inserting the expressions for s₁₁, s₂₂ and s₁₂, s₂₁ in equation 3,we obtain $\begin{matrix}\begin{matrix}{\Gamma_{a\quad c} \cong {\delta + \frac{{\Delta\quad Z} + {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}} +}} \\{{\left\lbrack {\mu - \frac{{\Delta\quad Z} - {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}}} \right\rbrack{\mathbb{e}}^{{- 4}{\gamma 1}}} -} \\{\left\lbrack \quad{1 + \tau + {\frac{\Delta\quad Z}{Z_{0}}{\sinh\left( {2\gamma\quad l_{2}} \right)}} - {j\frac{{\Delta\quad X_{2}} - {\Delta\quad Y_{2}Z_{0}^{2}}}{Z_{0}}{\cosh\left( {2\gamma\quad l_{2}} \right)}}} \right\rbrack{\mathbb{e}}^{{- 2}{\gamma 1}}}\end{matrix} & (8)\end{matrix}$With Re (2γ l₂)<<1, the simplified relationship $\begin{matrix}\begin{matrix}{\Gamma_{a\quad c} \cong {\delta + \frac{{\Delta\quad Z} + {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}} +}} \\{{\left\lbrack {\mu - \frac{{\Delta\quad Z} - {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}}} \right\rbrack{\mathbb{e}}^{{- 4}\gamma\quad l}} -} \\{\left\lbrack {1 + \tau + {j\frac{\Delta\quad Z}{Z_{0}}\sin\left( {2\beta\quad l_{2}} \right)} -} \right.} \\{\left. {j\frac{{{\Delta\quad X_{2}} - {\Delta\quad Y_{2}Z_{0}^{2}}}\quad}{Z_{0}}{\cos\left( {2\beta\quad l_{2}} \right)}} \right\rbrack{\mathbb{e}}^{2\gamma\quad l}}\end{matrix} & (9)\end{matrix}$applies.WithΔX₁=ωΔL₁   (10)ΔX₂=ωΔL₂   (11)ΔY₁=ωΔC₁   (12)ΔY₂=ωΔC₂   (13)the following applies: $\begin{matrix}{{{\mathbb{e}}^{{- 2}\gamma\quad l}\begin{matrix}{\Gamma_{a\quad c} \cong {\delta + \frac{{\Delta\quad Z} + {j\quad{\omega\left( {{\Delta\quad L_{1}} - {\Delta\quad C_{1}Z_{0}^{2}}} \right)}}}{2Z_{0}} +}} \\{{\left\lbrack {\mu - \frac{{\Delta\quad Z} - {j\quad{\omega\left( {{\Delta\quad L_{1}} - {\Delta\quad C_{1}Z_{0}^{2}}} \right)}}}{2Z_{0}}} \right\rbrack{\mathbb{e}}^{{- 4}\gamma\quad l}} -} \\{\left\lbrack {1 + \tau + {j\left\lbrack {{\frac{\Delta\quad Z}{Z_{0}}{\sin\left( {2\beta\quad l_{2}} \right)}} -} \right.}} \right.} \\\left. \left. {\omega\frac{{\Delta\quad L_{2}} - {\Delta\quad C_{2}Z_{0}^{2}}}{Z_{0}}{\cos\left( {2\beta\quad l_{2}} \right)}} \right\rbrack \right\rbrack\end{matrix}}{With}} & (14) \\{r_{z} = \frac{\Delta\quad Z}{2Z_{0}}} & (15) \\{t_{1} = \frac{{\Delta\quad L_{1}} - {\Delta\quad C_{1}Z_{0}^{2}}}{2Z_{0}}} & (16) \\{t_{2} = \frac{{\Delta\quad L_{2}} - {\Delta\quad C_{2}Z_{0}^{2}}}{2\quad Z_{0}}} & (17)\end{matrix}$we finally obtain:Γ_(ac) ≅δ+r ₂ +jωt ₁ +[μ−r _(z) +jωt ₁ ]e ^(−ayl)−[1+τ+j[2r _(z) sin(2βl ₂)−2ωt ₂ cos (2βl ₂)]]e ^(−2y)   (18).Γ_(ac) is a function of the frequency, and in the case of a linearsweep,ω=κt   (19)also a function of time. One may then imagine Γ_(ac) as the sum of acomplex oscillation with the carrierA=−[l+τ+j[2r _(z) sin (2βl ₂)−2ωt ₂ cos (2βl ₂)]]e ^(−2yl)   (20)a baseband signalB=δ+r _(z) +jωt ₁   (21)and a signal at the double carrier frequencyC=[μ−r _(z) +jωt ₁ ]e ^(−4yl)   (22),leading to the characteristic ripple of |Γ_(ac)| over the frequency. Aswill be shown below, the spectral portions of Γ_(ac) referred to may beobtained through a discrete Fourier transformation (DFT) and subsequentfiltering out, so that the sought variables δ and μ may be calculated asfollows: $\begin{matrix}{\mu = {{C\frac{\left\lbrack {1 + \tau + {j\left\lbrack {{2r_{2}{\sin\left( {2\beta\quad l_{2}} \right)}} - {2\quad\omega\quad t_{2}{\cos\left( {2\beta\quad l_{2}} \right)}}} \right\rbrack}} \right\rbrack^{2}}{A^{2}}} + r_{z} - {j\quad\omega\quad t_{1}}}} & (23)\end{matrix}$δ=B−(r _(z) +jωt ₁)   (24).

Equation 23 may be further simplified without much change in theaccuracy with which μ may be determined: $\begin{matrix}{\mu \equiv {\frac{C}{A^{2}} + {\left( {r_{z} - {j\quad\omega\quad t_{1}}} \right).}}} & (25)\end{matrix}$

The terms r_(z)±jωt₁ define an error vector which stems only from theair line used, and their value corresponds to the impedance deviation(including male connector influence). If this vector or at least themore easily determined variable r_(z) is available, then air lines withgreater impedance tolerance may also be used.

Correction of the error terms “system directivity” (directivity) D and“system port impedance match” (source port match) M of the VNA using thevalues obtained is effected by the following equations:D _(neu) =D _(alt) δT   (26)M _(neu) =M _(alt)μ  (27).Case 2: |Γ₁|<0.1 (Verification With Small Outlet-Side False Termination)

For this case, equation 3 may first of all be simplified:Γ_(ac) ≅δ+s ₁₁+(1+τ+s ₂₂Γ₁)s ₁₂ s ₂₁Γ₁   (28)

After inserting the s-parameter of the air line, we obtain$\begin{matrix}{\delta + \frac{{\Delta\quad Z} + {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}} - {\frac{{\Delta\quad Z} - {j\left( {{\Delta\quad X_{1}} - {\Delta\quad Y_{1}Z_{0}^{2}}} \right)}}{2Z_{0}}\Gamma_{1}^{2}{\mathbb{e}}^{{- 4}{rl}}} + {\left( {\Gamma_{1} + {\tau\Gamma}_{1} + {\left\lbrack {\frac{{\Delta\quad Z} + {j\quad\Delta\quad X_{2}}}{2Z_{0}} - {j\frac{\Delta\quad Y_{2}Z_{0}}{2}}} \right\rbrack\Gamma_{1}^{2}{\mathbb{e}}^{{- 2}{\gamma 1}_{2}}} - {\left\lbrack {\frac{{\Delta\quad Z} - {j\quad\Delta\quad X_{2}}}{2Z_{0}} + {j\frac{\Delta\quad Y_{2}Z_{0}}{2}}} \right\rbrack{\mathbb{e}}^{{- 2}\gamma\quad l_{2}}} -} \right){{\mathbb{e}}^{2\gamma\quad l}.}}} & (29)\end{matrix}$

The signal contains three spectral components—baseband (lower sideband),“carrier” (. . . xe^(−2yl)) and upper sideband (. . . xe^(−4yl))—ofwhich the upper sideband is not likely to be capable of evaluation owingto its small magnitude. The “carrier” contains no information ofinterest, and the baseband is identical to that obtained for theshort-circuited air line (term B in equation 21).

The evaluation of the measured results is explained below.

Case 1: Γ₁=−1 (Verification With Outlet-Side Short-Circuit)

Determination of the components B and C/A² from the measured results forΓ_(ac) is described below. Here it is assumed that N equidistantmeasuring points Γ_(nc) (n=0 . . . N-1) are available (FIGS. 4 and 5):

-   1) Fourier transformation with search for the dominating component F    (FIG. 6).-   2) Zero mixing of the carrier by multiplying Γ_(nc) by    $\begin{matrix}    {{{\mathbb{e}}^{{j2\pi}\quad F\frac{\pi}{N}}:\Gamma_{{ac},{mix}_{4}}} = {\Gamma_{{ac}_{n}} \cdot {\mathbb{e}}^{{j2}\quad\pi\quad F\frac{n}{N}}}} & (30)    \end{matrix}$-   Extension of the point sequence Γ_(ac, mix) my adding E_(e) points    at the end (n_(c)=N . . . N+E_(e)−1), which are obtained by    extrapolation with a linear predictor via the last P points (p=N−P-1    . . . N-1).-   4) Extension of the point sequence by adding E_(c) points at the    beginning (n_(a)=−E_(a) . . . −1), obtained by extrapolating the    reflected sequence {circumflex over (Γ)}_(ac,mix)=Γ_(ac,mix)(v=0 . .    . N+E_(c)−1) in the manner described under 3 above.-   5) The extrapolated reflected sequence of 4) with N+E_(a)+E_(c)    points is reflected back and the sequence of 4) added to it,    resulting in a new sequence Γ_(ac,mix) _(k) with k=2 (N+E_(a)+E_(e))    points. By definition the beginning and end points have the same    value (FIG. 7).-   6) Discrete Fourier transformation and obtaining of the mixed-down    carrier with an ideal low-pass filter (rectangular transfer    function, no delay time, see FIG. 8).-   7) Inverse Fourier transformation of the carrier and cutting out of    the intersecting section of    A′ _(n)(n=0 . . . N−1)-   8) Obtaining the carrier in the original position by mixing upwards    (FIG. 9): $\begin{matrix}    {A_{n} = {A_{n}^{\prime}{\mathbb{e}}^{{- {j2\pi}}\quad F\frac{n}{N}}}} & (31)    \end{matrix}$-   9) Subtraction of the carrier from Γ_(ac) _(n) , so that the    baseband signal and the signal for the doubled carrier sequence are    left over (FIG. 10):    S _(ac) _(n) =Γ_(ac) _(n) −A _(n)   (32)-   10) Obtaining the baseband signal (B_(n)) from S_(ac) (FIGS. 11,    12), as for 3) to 7).-   11) Obtaining the term C_(n)/A_(n) ² from S_(ac)/A² (FIGS. 13, 14,    15) as for 3) to 7).

FIG. 16 shows the result of the values thus calculated from the measuredreflection coefficients for the value of the effective directivity andthe value of the effective source port match, and specifically incomparison with the value of the ripple amplitude of the oscillationwhich is superimposed on the value of the reflection coefficients. FIG.16 shows that there are considerable differences between this rippleamplitude, formerly evaluated on its own, and the values measured inaccordance with the invention, with the values measured in accordancewith the invention and evaluated separately being significantly moreaccurate.

For the usability of the method according to the invention it is quiteimportant that the “spectra” of the components B and C do not overlapthat of the carrier A, i.e. that the distances of the reflectionscontained in B and C from the reference plane are less than the lengthof the air line. This in turn means that “residual system directivity”and “residual system port impedance match” should no longer contain anyportions of the physical network analyser, but only portions stemmingfrom the inadequacy of the calibration standards used. Consequently themethod according to the invention may be implemented only after systemcalibration of the VNA.

A quite major difficulty in obtaining spectral portions from an endlesssection of time arises from the fact that the Fourier transformationforms the spectrum of the periodically repeated signal section. Apartform the discretization in frequency terms, this gives rise to spectralcomponents which are not actually present in the signal. If now thesought portions are cut out of this distorted spectrum, then inevitablysome of the lines added by the periodic repetition are also lost, sothat the pattern over time after transformation is additionallydistorted.

There are two known methods of reducing this effect. On the one hand thesignal section may be made part of a window before the Fouriertransformation, so that the signal begins and ends close to zero andthus only minimal spectral distortion occurs. However it is not possibleto rely on the beginning and end coinciding after retransformation, i.e.the method does not function over the whole signal section, i.e. onlyover the range 8 GHz to 32 GHz for a set of measured values extendingover 100 MHz to 40 GHz.

In the second known method, the reflected pattern is once again attachedto the available signal section, which at least enforces a steadypattern and thus minimal spectral distortions of the periodicallyrepeated signal. Nevertheless here too the beginning and end of thesignal section may be used only with limitations after retransformation.

The method according to the invention also utilises the possibility ofreflection, but before that makes an extrapolation of the signal sectionover the end and the beginning (in the direction of “negative frequencyvalues”). For extrapolation, a linear predictor is used, which leads toa steady and—in terms of frequency continuity—differentiable pattern atthe end points. After doubling of the signal section through reflection,the discrete Fourier transformation (DFT) is effected, obtaining thedesired spectral portion and retransformation. The boundary sections,i.e. the extrapolated portions, which are in any case only slightlydistorted, are then cut off and the remaining signal is furtherprocessed.

Linear prediction allows the calculation of the extrapolated values as alinear combination of preceding values: $\begin{matrix}{\Gamma_{2c_{n}} = {{\sum\limits_{e = 1}^{E}\quad{a_{e}\quad{{Re}\left( \Gamma_{2c_{n - a}} \right)}}} + {j\quad{\sum\limits_{e = 1}^{E}\quad{b_{e}\quad{{{Im}\left( \Gamma_{2c_{n - a}} \right)}.}}}}}} & (33)\end{matrix}$

Here the weights a_(c) and b_(e) are so determined that the applicationof equation (33) within the measured signal section leads to thesmallest possible errors. The mathematical processes for itsdetermination are described in the relevant literature. It has beenfound that an extension of the measured signal section to double thevalue is normally sufficient for the purposes of the invention. Byadding even more points, and if possible up to a local extreme of thevalue (FIG. 18), on the one hand the frequency resolution is furtherenhanced, and on the other hand the scope for enhanced selectivity ofthe method is created (FIG. 17). The latter is important when thebaseband and the frequency band for the doubled carrier frequency arerelatively broad and come close to the carrier (generated by reflectionswhich are relative far removed from the reference plane).

The sought spectral components are obtained by low-pass filtering andsubsequent retransformation. It is quite satisfactory to use “ideal” lowpasses with rectangular transfer function (1 in the passband range, 0 inthe attenuation band), with a bandwidth chosen so that the transfer fromthe passband to the attenuation band lies in the middle between the twospectral components. Any restriction of the bandwidth below this valueleads on the one hand to a (desired) reduction of the noise overlyingthe measured result, but may on the other hand lead to distortions ofthe measured result, if relevant spectral portions are eliminated bythis action. In practice it is necessary to find a compromise, in whichthe choice of bandwidth for the measuring task described might be ratedas non-critical (FIGS. 14 and 15). Even with extreme reduction (FIG. 12right), reasonable measured results could be obtained. If it isremembered that the preceding system calibration was carried out by theOSM method (calibration standards Open, Short and Match), and that asliding load from 4 GHz upwards was used, then the result is also easilyexplained: since the air line section of sliding load and reference airline differ mainly due to their characteristic (frequency-independentand real) impedance, and to the reactances of the plug connections atthe reference plane, which rise in proportion to frequency, a curve suchas that of FIG. 12 (right) is also to be expected in purely theoreticalterms.

Case 2: |Γ₁|<0.1 (Verification With Small Outlet-Side False Termination)

The evaluation may be made under precisely the same procedure as for theverification with short-circuit (item 11 is omitted). The result foreffective directivity (“residual system directivity”) should beidentical to that obtained for the verification with short-circuit. Norequirements are set for the magnitude of the false termination relativeto the effective directivity (in contrast to the ripple evaluationaccording to the prior art).

Discrete Fourier transformation (DFT) of the extended and reflectedsequence of reflection measured values actually produces a time signalcomparable to the pulse response of the system. If nevertheless the term“spectrum” has been used for this purpose above, then this is onlybecause the associated terms and concepts are more familiar.Incidentally, it is possible through equation (19) to imagine thereflection coefficients as a time signal, so that the term “spectrum”for the DFT of the reflection coefficients has its justification. Themathematical algorithms used are in any case to be used irrespective ofthe terminology.

1. Method of measuring the effective directivity of a test port of asystem-calibrated vector network analyser by the steps of: connecting aprecision air line having an inlet and an outlet and short-circuited atthe outlet end; measuring complex reflection coefficients at the inletof the precision air line at a sequence of measuring points within aprescribed frequency range; subjecting the sequence of the measuredcomplex reflection coefficients to a discrete Fourier transformation toform a spectrum; filtering out the baseband from the spectrum; andobtaining a sequence of effective directivity values by subsequentinverse Fourier retransformation.
 2. Method of measuring the effectivesource port match of a test port of a system calibrated vector networkanalyser by the steps of: connecting a precision air line having aninlet and an outlet and short-circuited at the outlet; measuring complexreflection coefficients at the inlet of the precision air line at asequence of measuring points within a prescribed frequency range;subjecting the sequence of the measured complex reflection coefficientsto a complex Fourier transformation to form a spectrum; filtering outthe dominant component (carrier) from the spectrum; determining the sizeof this dominant component by inverse Fourier retransformation; thenfiltering out an upper sideband at double the frequency of the dominantcomponent; determining the size of the filtered-out upper sideband byinverse Fourier retransformation; and finally, obtaining a sequence ofeffective source port match value by respective division of the variableof the dominant component (carrier) by the square of the variable of theupper filtered-out sideband.
 3. Method of measuring the effectivedirectivity and the effective source port match of a test port of asystem-calibrated vector network analyser by the steps of: (a)connecting a precision air line having an inlet and an outlet andshort-circuited at the outlet; (b) measuring complex reflectioncoefficients at the inlet of the precision air line at a sequence ofmeasuring points within a prescribed frequency range; (c) subjecting thesequence of the measured reflection coefficients to a discrete Fouriertransformation to form a spectrum; (d) filtering out the baseband andthe dominant component (carrier) from the spectrum; (e) determining thesize of this dominant component, and; (f) obtaining a sequence ofeffective directivity values by inverse Fourier transformation. 4.Method according to claim 1, wherein the precision air line has animpedance deviating from a system impedance of the network analyser by aknown value, the method comprising taking the known impedance deviationinto account as appropriate correction values in determining thesequence of values of the effective directivity and/or effective sourceport match.
 5. Method according to claim 1, comprising the steps ofextrapolating the sequence of the measure complex reflectioncoefficients by linear prediction before the complex Fouriertransformation of the sequence of the measured complex reflectioncoefficients, and extending the original sequence by reflecting thisextrapolated sequence of complex reflection coefficients.
 6. Methodaccording to claim 5, comprising the step of multiplying the sequence ofcomplex reflection coefficients by a factor such that the dominatingcomponent (carrier) lies at the frequency zero due to downwards mixingin the spectrum before the linear prediction.
 7. Method according toclaim 1, comprising the steps of connecting an air line with a slightfalse termination at the outlet (|Γ_(e)|0.1) to the test port to bemeasured of the system-calibrated vector network analyser, and thentreating the measured complex reflection coefficients in accordance withthe steps of claim
 1. 8. Method according to claim 1, comprising thestep of connecting system error terms of the network analyser by theobtained values of the effective directivity.
 9. Method according toclaim 1, comprising the step of storing the obtained sequence of valuesof the effective directivity on a data medium which may be driventogether with the calibration standards used for system calibration ofthe network analyser.
 10. Set of calibration standards for systemcalibration of vector network analysers with various calibrationstandards for open-circuit, short-circuit and match, assigned a datamedium with values of the effective directivity, which have beenobtained according to the method set out in claim
 1. 11. Methodaccording to claim 1, comprising the step of storing the obtainedsequence of values of the effective directivity on a data medium whichmay be driven together with the calibration standards used for systemcalibration of the network analyser:
 12. Set of calibration standardsfor system calibration of vector network analysers with variouscalibration standards for open-circuit, short-circuit and match,assigned a data medium with values of the effective directivity whichhave been obtained according to the method set out in claim 1.